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Basic Algebraic Geometry and the 27 Lines on a Cubic Surface

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Basic Algebraic Geometry and the 27 Lines on a Cubic Surface BASIC ALGEBRAIC GEOMETRY AND THE 27 LINES ON A CUBIC SURFACE SIMON LAZARUS Abstract. We present an elementary proof of the fact that any nonsingular 3 cubic surface in P over an algebraically closed field not of characteristic 2 will contain exactly 27 lines. During the course of this proof, we present a way to label these lines in terms of a fixed line l and another fixed line m which is disjoint from l in such a way that we can determine which of the 26 other lines a given line will intersect. In order to present the proof, we first develop without proof some of the basic theory of algebraic geometry and also prove 3 several facts about quadric surfaces and lines in P . Contents 1. Preliminaries 1 1.1. Affine Algebraic Varieties 1 1.2. Projective Algebraic Varieties 4 1.3. Irreducibility and Nonsingularity 7 2. Miscellaneous Lemmas 9 3. The 27 Lines on a Nonsingular Cubic Surface 14 Acknowledgments 23 References 23 1. Preliminaries Throughout this paper, let K be a field. We shall mostly be interested in the case where K is algebraically closed and not of characteristic 2, but to begin we can consider any field K. 1.1. Affine Algebraic Varieties. Definition 1.1. Let C be any collection of polynomials in K[x1; : : : ; xn]. Define the affine algebraic variety associated to C to be V(C) = fp 2 Kn : 8f 2 C; f(p) = 0g; the common zero set of all polynomials in C. More generally, an affine algebraic variety is any subset of Kn of the form V(D) for some collection D of polynomials. For the sake of brevity, we shall sometimes use the term affine variety or simply variety to describe an affine algebraic variety. Intuitively, an affine algebraic variety is a subset of Kn which is \cut out" from Kn by some collection of polynomials. Date: September 16, 2014. 1 2 SIMON LAZARUS It is easy to check that the finite union or arbitrary intersection of affine algebraic varieties is an affine algebraic variety. Specifically, the union of two varieties is given by (1.1) V(ffigi2I ) [V(fgjgj2J ) = V(ffigjgi2I;j2J ) and the intersection of arbitrarily many varieties associated to collections Ci of polynomials is given by ! \ [ (1.2) V(Ci) = V Ci : i2I i2I Additionally, the sets Kn = V(;) and ; = V(1), where 1 denotes the constant poly- nomial with value 1, are clearly affine algebraic varieties. Thus, we can naturally place a topology on Kn. Definition 1.2. The Zariski topology on Kn is the topology whose closed sets are n n n the affine algebraic varieties in K . Under this topology, we denote K by AK , or simply An when the underlying field K is clear. An is called Affine n-space. Any affine algebraic variety V is a subset of An and therefore inherits the sub- space topology from An. Under this topology, the closed sets of V are precisely the subvarieties of V , i.e. any subset of V which is the common zero set of some collection of polynomials on V . We now describe a correspondence between varieties and ideals in the polynomial ring K[x1; : : : ; xn] which is of great importance. Definition 1.3. Let V ⊆ An be an affine algebraic variety. Define the ideal of V to be I(V ) = ff 2 K[x1; : : : ; xn]: 8p 2 V; f(p) = 0g; the set of polynomials vanishing on V . The set I(V ) is clearly an ideal of K[x1; : : : ; xn]: the sum of polynomials van- ishing on V also vanishes on V , and the product of a polynomial vanishing on V with any other polynomial must still vanish on V . Additionally, by definition I(V ) is alwaysp a radical ideal. (Recall that the radical of an ideal I in a ring R is defined n + to bep I = fx 2 R : x 2 I for some n 2 Z g and that I is said to be radical if I = I.) We now observe that if V is any affine algebraic variety, then V = V(I(V )). For first, V ⊆ V(I(V )) is obvious. Second, let p 2 V(I(V )). Then by definition for each f 2 I(V ) we have that f(p) = 0. So writing V = V(ffigi2I ), we certainly have that fi(p) = 0 for each i 2 I since necessarily each fi vanishes on V and hence is in I(V ). That is, p 2 V(ffigi2I ) = V . We might now similarly expect that if I is any radical ideal of K[x1; : : : ; xn] then I = I(V(I)). As it turns out, this is not always the case: for example, if K = R and I = (x2 + 1) then we have that V(I) = ; and hence I(V(I)) = R[x] 6= (x2 + 1). However, if K is algebraically closed, then our conjecture is in fact true: Hilbert's Nullstellensatz. Suppose K is algebraicallyp closed. Then for each ideal I of K[x1; : : : ; xn] we have that I(V(I)) = I. Thus, if I is radical then we have that I(V(I)) = I. BASIC ALGEBRAIC GEOMETRY AND THE 27 LINES ON A CUBIC SURFACE 3 For a proof, see [1]. Note that by definition, if V is any affine algebraic variety then I(V ) is radical. Thus, when combined with the above observation that for any variety V we have V(I(V )) = V , the Nullstellensatz gives us a bijective cor- respondence between the set A of radical ideals in K[x1; : : : ; xn] and the set B of affine algebraic varieties in An. For we have functions V : A ! B and I : B ! A which are inverse to each other. We now note that every affine algebraic variety can in fact be written as V(C) for some finite collection C of polynomials. This follows from the following theorem, whose proof also appears in [1]. Hilbert's Basis Theorem. Let R be a Noetherian ring. Then the polynomial ring R[x] is also Noetherian. Inductively, we see that if R is Noetherian, then R[x1; : : : ; xn] is Noetherian. Since any field is trivially Noetherian, we thus have that K[x1; : : : ; xn] is always Noetherian. So letting V be any affine algebraic variety in An, we see that the ideal I(V ) must be finitely generated; say I(V ) = (f1; : : : ; fk). Therefore V = V(I(V )) = V((f1; : : : ; fk)) = V(f1; : : : ; fk); where the last equality follows from the fact that if f1; : : : ; fk all vanish at a point n p 2 A then every polynomial in the ideal generated by f1; : : : ; fk must also vanish at p. Having defined our objects of interest, namely affine algebraic varieties, we now describe their morphisms. Definition 1.4. Let V ⊆ Am;W ⊆ An be affine algebraic varieties. A morphism of affine algebraic varieties is a function F : V ! W which is the restriction to V of some polynomial map G : Am ! An (that is, the restriction of a map G = (g1; : : : ; gn) where each gi 2 K[x1; : : : ; xm]). Such a function is an isomorphism if it has an inverse F −1 : W ! V which is also a morphism of affine algebraic varieties. Since an affine algebraic variety is by definition simply the common zero set of some collection C of polynomials, we would expect polynomial maps to be the functions which preserve the structure in which we are interested. However, we cannot make this intuition rigorous under out initial notion of a variety as simply being a set corresponding to a collection of polynomials. For example, consider 1 2 1 the varieties V1 ⊆ A and V2 ⊆ A given by V1 = A = V(;) and V2 = f(x; 0) : x 2 Kg = V(x2). Clearly projection onto the first coordinate is an isomorphism of affine algebraic varieties from V2 to V1. However, no collection of polynomials 1 cutting out V1 from A can ever be equivalent to a collection of polynomials cutting 2 1 out V2 from A : the only collections cutting out V1 from A are the empty collection and the collection f0g of the zero polynomial, while any collection cutting out V2 2 from A must generate an ideal containing the nontrivial polynomial x2. That is, although the two varieties V1 and V2 are isomorphic under our proposed definition of isomorphism, there is no way to make their defining collections of polynomials equivalent. To fix this problem, we introduce the following notion. Definition 1.5. Let V ⊆ An be an affine algebraic variety. The coordinate ring of V is defined to be K[V ] = K[x1; : : : ; xn]=I(V ): 4 SIMON LAZARUS The coordinate ring of V is simply the collection of polynomials on V . For in K[x1; : : : ; xn]=I(V ), two polynomials f; g 2 K[x1; : : : ; xn] (that is, two polynomials on An) are equivalent precisely when their difference vanishes on V . That is, two n polynomials on A are equivalent in K[x1; : : : ; xn]=I(V ) precisely when the two polynomials are the same on V . We note immediately that in our example above of the isomorphic varieties V1 and V2, we have that I(V1) = f0g and I(V2) = (x2). Thus, the coordinate ring of ∼ ∼ V1 is K[x1]=f0g = K[x1], and the coordinate ring of V2 is K[x1; x2]=(x2) = K[x1]. That is, the coordinate rings of V1 and V2 are isomorphic. As it turns out, there is an equivalence of categories between the set of affine algebraic varieties and the set of finitely generated K-algebras without nilpotent elements.
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